from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6171, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,8,25]))
pari: [g,chi] = znchar(Mod(4921,6171))
Basic properties
| Modulus: | \(6171\) | |
| Conductor: | \(187\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{187}(59,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6171.bq
\(\chi_{6171}(202,\cdot)\) \(\chi_{6171}(1600,\cdot)\) \(\chi_{6171}(1963,\cdot)\) \(\chi_{6171}(2671,\cdot)\) \(\chi_{6171}(3028,\cdot)\) \(\chi_{6171}(3034,\cdot)\) \(\chi_{6171}(3052,\cdot)\) \(\chi_{6171}(3391,\cdot)\) \(\chi_{6171}(3415,\cdot)\) \(\chi_{6171}(4123,\cdot)\) \(\chi_{6171}(4480,\cdot)\) \(\chi_{6171}(4486,\cdot)\) \(\chi_{6171}(4558,\cdot)\) \(\chi_{6171}(4843,\cdot)\) \(\chi_{6171}(4921,\cdot)\) \(\chi_{6171}(6010,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{40})\) |
| Fixed field: | 40.40.24562817038400928776197921239227357886542077974183334844678041435576602047153.1 |
Values on generators
\((4115,970,2179)\) → \((1,e\left(\frac{1}{5}\right),e\left(\frac{5}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(19\) |
| \( \chi_{ 6171 }(4921, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) |
sage: chi.jacobi_sum(n)